用AI发现组合数学中的路径映射算法，可人工验证。
Discovering a Zeta Map Algorithm on Dyck Paths via Mechanistic Interpretability
- 通过分析小模型的注意力机制，发现层级结构计算路径。
- 提出新算法，与经典映射一致，仅标签顺序不同。
- 适合对机器学习辅助数学发现感兴趣的组合学家。
机器学习在数学发现中的应用日益广泛，但数学更关注可独立验证的显式构造而非预测结果。本文以德克路径上的zeta映射为例，研究这一场景——这是q,t-Catalan数组合学中的经典双射。我们训练了一个小型单层、单头编码器-解码器Transformer，并利用机械可解释性工具（包括解码器交叉注意力分析、线性探测和因果干预）分析其学习到的计算过程。分析揭示了一种基于层级的机制：编码器表示使路径层级线性可访问，解码器则以结构化方式选择并遍历输入位置。将这些信号转化为组合数学，得到‘支架映射’——一种基于峰值中心的德克路径遍历算法。我们证明该算法与zeta映射一致，仅标签反转约定略有差异。本研究为人工智能辅助数学发现提供了可控范例，机械可解释性成功将模型行为转化为精确、可人工验证的组合算法。
原文摘要 · Abstract (English)
Machine learning is increasingly used in mathematical discovery, but in mathematics the desired output is often not a prediction itself, but an explicit construction that can be checked independently. We study this setting through the zeta map on Dyck paths, a classical bijection in the combinatorics of the q,t-Catalan numbers. We train a deliberately small one-layer, one-head encoder-decoder transformer on this map and analyze its learned computation using mechanistic interpretability tools, including decoder cross-attention analysis, linear probing, and causal intervention. The analysis reveals a level-based mechanism: encoder representations make path levels linearly accessible, while the decoder selects and traverses input positions in a structured way. Translating these signals into combinatorics leads to the scaffolding map, an explicit peak-centered traversal algorithm for Dyck paths. We prove that this algorithm agrees with the zeta map, modulo a reversal convention in the labeling. This gives a controlled example of AI-assisted mathematical discovery in which mechanistic interpretability turns model behavior into a precise, human-verifiable combinatorial algorithm.
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